Summary: Quantum Versus Classical Proofs and Advice
Scott Aaronson
Greg Kuperberg
Abstract
This paper studies whether quantum proofs are more powerful than classical proofs, or in complexity terms,
whether QMA = QCMA. We prove three results about this question. First, we give a "quantum oracle separation"
between QMA and QCMA. More concretely, we show that any quantum algorithm needs
"q
2n
m+1
"
queries
to find an n-qubit "marked state" | , even if given an m-bit classical description of | together with a quantum
black box that recognizes | . Second, we give an explicit QCMA protocol that nearly achieves this lower bound.
Third, we show that, in the one previously-known case where quantum proofs seemed to provide an exponential
advantage, classical proofs are basically just as powerful. In particular, Watrous gave a QMA protocol for verifying
non-membership in finite groups. Under plausible group-theoretic assumptions, we give a QCMA protocol for the
same problem. Even with no assumptions, our protocol makes only polynomially many queries to the group oracle.
We end with some conjectures about quantum versus classical oracles, and about the possibility of a classical oracle
separation between QMA and QCMA.